Variational principles for differential equations with symmetries and conservation laws. I: Second order scalar equations
The Noether correspondence between the symmetries of a variational problem and conservation laws for the Euler-Lagrange equations suggests the following general problem. Suppose a system of differential equations admits a large group of symmetries and that each one of these symmetries generates a conservation law for the equations. Must these differential equations be the Euler-Lagrange equations for some problem in the calculus of variations? In 1977, F. Takens formulated this problem rigorously and answered it affirmatively in several special cases. In this paper and its sequel we generalize a number of Takens' results. Here we undertake a detailed study of this problem for the case of second order scalar differential equations. Let \(\Gamma\) be a given infinitesimal Lie transformation group acting on the space of independent and dependent variables. We show that if a second order scalar differential equation \(\Delta=0\) is \(\Gamma\) invariant and admits \(\Gamma\) conservation laws, then the Helmholtz operator associated to \(\Delta\) must satisfy certain differential conditions. A careful analysis of these differential equations allows us to substantially generalize Takens' result for second order scalar differential equations. For a particular class of infinitesimal Lie groups \(\Gamma\), we are able to completely reduce the solution of the differential conditions on the Helmholtz operator to the solution of a simple system of equations depending only on the structure constants of \(\Gamma\). We illustrate the interplay between symmetries, conservation laws and variational principles for second order scalar partial differential equations with many examples.
second order scalar differential equations
scientific article; zbMATH DE number 683140
scientific article